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# CM Elliptic Curves: Volcanoes, Reality and Applications, Part II

Dec 2022

Let $M \mid N$ be positive integers, and let $\Delta$ be the discriminant ofan order in an imaginary quadratic field $K$. When $\Delta_K < -4$, the firstauthor determined the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ overthe closed point $J_{\Delta}$ corresponding to $\Delta$ and showed that allfibers of the map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_{\Delta}$ wereconnected. Here we complement this prior work by addressing the most difficultcases $\Delta_K \in \{-3,-4\}$. These works provide all the information neededto compute, for each positive integer $d$, all subgroups of$E(F)[\operatorname{tors}]$, where $F$ is a number field of degree $d$ and$E_{/F}$ is an elliptic curve with complex multiplication.

Q1论文试图解决什么问题？
Q2这是否是一个新的问题？
Q3这篇文章要验证一个什么科学假设？
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